For \(n \in N\), if \(I_n=\int \frac{\sin n x}{\sin x} d x=\frac{2}{n-1} \sin (n-1) x+I_{n-2}\) and \(\int_0^\pi \frac{\sin n x}{\sin x} d x=\frac{k \pi}{2}\), then \(k=\)

If \(\sqrt{\frac{y}{x}}+\sqrt{\frac{x}{y}}=2\), then \(\frac{d y}{d x}\) is equal to

\(\lim _{n \rightarrow \infty} \frac{1}{n}\left(\frac{1}{e^{1 / n}}+\frac{1}{e^{2 / n}}+\frac{1}{e^{3 / n}}+\ldots+\frac{1}{e^2}\right)=\)

If \(A=\left[\begin{array}{ccc}1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1\end{array}\right]\), then \(A^{-1}=\)

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

If one root of the equation $i{x}^2-2(i+1)x+(2-i)=0$ is $(2-i)$, then the other root is

If \(\left|\begin{array}{lll}a & a^3 & a^4 \\ b & b^3 & b^4 \\ c & c^3 & c^4\end{array}\right|=k(a-b)(b-c)(c-a)\) then \(\mathrm{k}=\)

If the acute angle between the pair of tangents drawn from the origin to the circle \(x^2+y^2-4 x-8 y+4=0\) is \(\alpha\), then \(\tan \alpha=\)

If the sum of two roots of the equation \(x^3-2 p x^2+3 q x-4 r=0\) is zero, then the value of \(r\) is

When a force \(\mathbf{F}\) given by \(\mathbf{F}=(6 \hat{\mathbf{i}}-18 \hat{\mathbf{j}}+10 \hat{\mathbf{k}}) \mathrm{N}\) acts on a body, it imparts an acceleration of \(8 \mathrm{~ms}^{-2}\). Then, find the mass of the body.

\(\int_1^{e^2} \frac{d x}{x(1+\log x)^2}=\)

An ideal gas \((\mathrm{X})\) present in a vessel of volume \(\mathrm{V} \mathrm{L}\) exerted a pressure of \(16.4 \mathrm{~atm}\) at \(200 \mathrm{~K}\). What is its concentration in \(\mathrm{mol} \mathrm{L}^{-1}\) ?
(Given \(\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) )

The solution of the differential equation \((2 x-4 y+3) \frac{d y}{d x}+(x-2 y+1)=0\) is
( \(C\) is an arbitrary constant)